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Article 5969 of comp.ai.philosophy:
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>From: costello@CS.Stanford.EDU (T Costello)
Newsgroups: comp.ai.philosophy
Subject: Re: penrose
Message-ID: <1992May29.012700.7102@CSD-NewsHost.Stanford.EDU>
Date: 29 May 92 01:27:00 GMT
Article-I.D.: CSD-News.1992May29.012700.7102
References: <atten.706786286@groucho.phil.ruu.nl> <1992May27.115843.13837@sics.se>
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In article <1992May27.115843.13837@sics.se>, torkel@sics.se (Torkel Franzen) writes:
|> In article <1992May27.105042.29890@CSD-NewsHost.Stanford.EDU> costello@CS.
|> 
|>   Your axioms have nothing in particular to do with the informal
|> reflection principle I formulated. To repeat, that principle was
|> "if a theory T is sound, any extension by reflection of T is also
|> sound". Here "extension by reflection" is itself not a formal concept,
|> but refers to various reflection axioms that may be added. The basic
|> formal reflection principle by which any sound theory T may be soundly
|> extended consists in the axiom schema
|> 
|>           BevT(+A+) -> A
|> 
|> for every sentence A, where +A+ is a name of the statement A and BevT a
|> suitably defined provability predicate for T. I don't pretend to understand
|> any notion "A is provable", where "provable" does not mean provable in a
|> specified theory.

The last three axioms I gave are an example of a system where it is inconsistent
to add your reflection principle.  This corresponds to Thompson's analysis of 
Belief.  Again if we coould have
Bew(+Bew(+A+) \supset A+) 
Bew(+logical axiom+)
Bew(+X+) \land Bew+X \supset y+) \supset Bew(+Y+)
Where Bew is the arithmetic predicate that states there is a proof of the
statement whose Godel number it is given, from the axioms of logic, and the 
axioms Bew(+A+) \supset A. The third axiom  above is a consequence of it
deinition.  Your informal reflection principle causes the system to
be inconsistent.



|> 
|>    >The notion of progression is not that complicated in Fefermann's paper.
|>    >It merely consists of adding axioms for each ordinal.  The system itself
|>    >acts as one might suspect.  The strangeness of the result ( in the light
|>    >of Godel) does not come from any magic in the progression.
|> 
|>   Your description is inadequate. Recursive progressions are much more
|> complicated than transfinite sequences of theories, although the
|> latter are rather more interesting from a philosophical point of view.
|> Feferman notes that "with regard to questions of completeness for
|> transfinite sequences of theories, these results must be treated with
|> care. Because of the intensional character of the construction of
|> recursive progressions [the theories associated with ordinal notations
|> d and d' may be different, even though d and d' denote the same
|> ordinal]."

In all of these discussion we have been considering sets of axioms and inference rules
rather than theories, and thus are treating the intensional rather than the extensional.

|> 
|>   What strikes you as strange about the result?

The strange result is of course that, while the notion of a recursive progression
seems meaningful, we cannot have a autonomous progression, because of the non elementary
nature of the predicate we would have to apply induction to to express this sequence.

Tom


